# How do you solve x/2 = sqrt( x - 1)?

Aug 19, 2015

$x = 2$

#### Explanation:

First, start by taking a look at the equation

$\sqrt{x - 1} = \frac{x}{2}$

Right from the start, you need $x$ to be positive. Not only that, but you need the expression that's under the square root to be positive as well, since you cannot take the square root of a negative number and get a real number as a solution.

So, you know that you need

$x - 1 \ge 0 \implies x \ge 1$

Next, square both sides of the equation to get rid of the square root

${\left(\sqrt{x - 1}\right)}^{2} = {\left(\frac{x}{2}\right)}^{2}$

$x - 1 = {x}^{2} / 4$

This is equivalent to

${x}^{2} - 4 x + 4 = 0$

Use the quadratic formula to find the solutions to this quadratic equation

${x}_{1 , 2} = \frac{- \left(- 4\right) \pm \sqrt{{\left(- 4\right)}^{2} - 4 \cdot 1 \cdot \left(- 4\right)}}{2 \cdot 1}$

${x}_{1 , 2} = \frac{4 \pm \sqrt{0}}{2}$

This means that the quadratic has one distinct solution

$x = \frac{4}{2} = \textcolor{g r e e n}{2}$

Since $x = 2$ satisfies the condtion $x \ge 1$, this will also be the solution to your original equation.

You can do a quick check to make sure that the calculations are correct

$\sqrt{2 - 1} = \frac{2}{2}$

$\sqrt{1} = 1$

$1 = 1 \text{ } \textcolor{g r e e n}{\sqrt{}}$